Haldane-Holstein model at fractional filling: Route to bosonic fractional Chern insulator and quantum anomalous Hall crystal

Electron-phonon coupling is generally expected to suppress band topology, driving a topological band insulator into a trivial phase. Here we show that, using the prototypical Haldane-Holstein model at partial filling of a topological band, strong electron-phonon coupling can instead stabilize topological phases of matter. The electron-phonon coupling plays a double role: it generates the longer-range interactions that correlate the carriers and, in the attractive channel, also binds them into bosonic pairs. For spinful electrons, it pairs opposite spins into charge-$2e$ bipolarons that form a bosonic fractional Chern insulator (FCI) at filling $\nu =1\mathrm{/}2$ with charge-$e$ semionic excitations. Tuning the band topology and the coupling strength maps out a rich phase diagram containing this bosonic FCI, superconductors condensed at the $M$ and $K$ points, and several charge-ordered solids. For spin-polarized electrons, strong coupling instead stabilizes a $C=2$ quantum anomalous Hall crystal (QAHC) at $\nu =1\mathrm{/}3$. Electron-phonon coupling thus emerges as a route to, rather than an obstruction against, topology in partially filled Chern bands.